Log of both examples is like 1/x, which satisfies the negative derivative condition. What am I missing that makes them counterexamples? Gerhard "Ask Me About System Design" Paseman, 2012.01.25
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Gerhard PasemanJan 25 '12 at 19:36

Both functions are log-convex, not log-concave.
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Mark MeckesJan 25 '12 at 20:01

So then when the original poster says (log f)' < 0, they mean f is log convex? Gerhard "Sometimes Confuses Up And Down" Paseman, 2012.01.25
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Gerhard PasemanJan 25 '12 at 21:07

The poster said (log f)''<0 (second derivative, not first). Although that's apparently the opposite of what was meant.
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Mark MeckesJan 26 '12 at 15:10

Actually, I meant log-convex instead of log-concave. I missed a minus sign in my derivations and this led to the conjecture that log(f) should be concave. I have found a more complete answer in this paper:

Theorem 1 establishes a series of inequalities for the derivatives of c.m. functions. In particular, by taking n=0 and m=1 in 3.2 we get the log-convexity of f.
I suggest to read this paper because of the relrevance of Theorem 1.